The lecture hall at the Institute for Advanced Study in Princeton resembled a crucible of thought; beneath its dome gathered mathematical elites from around the world. The air carried a peculiar intensity—an almost suffocating intellectual density produced by countless top minds operating simultaneously. Today, the seats were unusually packed; even corridors and rear floor space were filled. What drew them wasn't merely Yue'er's groundbreaking progress in recent years at the intersection of the P vs. NP problem and the Langlands program, but also her plan to publicly outline, for the first time in this talk, an even grander vision aiming to unify different mathematical fields.
Yue'er stood at the podium, adjusting the microphone. Dressed in a simple dark suit, her expression was calm; only those deep eyes held a brilliance as complex and bright as the mathematical structures she studied. On the large screen behind her appeared the talk's title: "From Computational Complexity to Mathematical Unity: Geometric Realization of Langlands Correspondence and Its Potential Implications." The title alone was enough to stir waves in the mathematical community.
She didn't rush into intricate formulas but began her speech with an almost poetic introduction.
"In early human civilization," Yue'er's voice, clear and steady, filled the quiet hall through the microphone, "different tribes, different cultures developed their own distinct languages and music. A tribe's priest used some ancient chant to pray for a bountiful harvest; musicians in a distant land used complex instrumentation to depict the motion of stars. Superficially, these chants and compositions seemed unrelated, belonging to different cultural lineages. But if we look deep enough, we might discover they were all attempting to express the same reverence for natural laws, the same quest for cosmic order. They use different 'musical scores,' yet may be recording different movements of the same 'cosmic symphony.'"
She paused, letting the metaphor settle in the listeners' minds.
"In the world of mathematics, we face a similar panorama. We have two seemingly disparate 'musical systems': one is **number theory**, which studies properties of integers, distribution of primes, solutions to equations—like the purest, most abstract primal chant; the other is **harmonic analysis**, which studies how functions decompose into more basic oscillatory components (such as Fourier analysis), focusing on spectral theory on symmetric spaces—like complex, layered symphonic music."
"For a long time, these two fields developed independently, each with its own language, methodology, and unsolved puzzles. They flowed like two parallel rivers, nourishing different territories of mathematics, yet seemingly never intersecting."
"However," Yue'er pivoted, her gaze sweeping over the attentive faces below, "Professor Robert Langlands, more than half a century ago, proposed an extraordinarily bold and profound conjecture—now known as the **Langlands program**. This program predicts that these two rivers, at some profoundly deep level, are connected! Certain core objects in number theory can establish one‑to‑one correspondences with certain core objects in harmonic analysis."
The screen behind her changed: on the left appeared the label "Number Theory," listing terms like "Galois representations," "L‑functions"; on the right appeared "Harmonic Analysis," listing "automorphic forms," "automorphic representations." In the center was a large question mark and a bidirectional arrow.
"The heart of the Langlands program is an attempt to construct a grand bridge at this question mark." Yue'er pointed to the bidirectional arrow. "This bridge allows us to 'translate' number‑theoretic problems into harmonic‑analysis problems to study, and vice versa. Sometimes a problem that's extremely difficult, nearly intractable in number theory may become clear or even solvable after translation to harmonic analysis; similarly, profound structures in harmonic analysis may in turn reveal hidden symmetries in number theory."
To help everyone grasp the essence of this "translation," she returned to the musical metaphor.
"Let's imagine," Yue'er's voice held a guiding magic, "one of the core objects in number theory, say a **Galois representation**, is like a 'mathematical score A' written in some extremely ancient, abstract cipher notation. This score records deep symmetric information about numbers and equations, but its notation is extremely obscure, hard to read and perform directly."
"Meanwhile, in harmonic analysis, one of its core objects, say an **automorphic form**, is like another 'mathematical score B' written in a completely different musical language (say modern staff notation). This score describes vibrational patterns of functions under the action of symmetry groups; its melodic and harmonic rules are relatively more familiar to mathematicians."
"**Langlands correspondence**," she emphasized, "essentially says that for each specific 'ancient‑cipher score A' (Galois representation), there exists a unique corresponding 'modern‑staff score B' (automorphic form); these two scores, despite their vastly different notations, describe **the same cosmic music**—the same mathematical reality expressed in different languages!"
This metaphor, like a flash of lightning, dispelled the fog surrounding the Langlands program in many non‑specialists' minds. Relating a highly abstract number‑theoretic structure to a relatively "concrete" analytic object was itself a stunning unification.
"My work," Yue'er continued, a trace of barely perceptible excitement in her tone, "is an attempt to find a **geometric realization** for this mysterious 'score‑translation' process." The screen changed again, showing diagrams of complex manifolds, varieties, and fiber bundles.
"I aim to show that we can simultaneously 'see' both scores within a carefully constructed **geometric space**. The structure of this geometric space itself encodes the translation rules. The symmetries of a **Galois representation** can manifest as symmetries of some 'covering' or 'fibration' structure on this space; the corresponding **automorphic form** can manifest as some vibrational pattern—defined on this space and satisfying a specific differential equation (e.g., eigenfunctions related to the Laplacian)."
She began introducing more concrete mathematical concepts, explaining how she used tools from algebraic geometry to construct deformations of special "Shimura varieties" and studied the structure of certain "ℓ‑adic cohomology" groups on them. These cohomology groups carry rich symmetry information, decomposable into irreducible components.
"In the cohomology of these geometric objects," Yue'er pointed to a key defining equation on screen, "on one hand we can identify structure arising from the number‑theory side—the action of the **Galois group**—corresponding to that 'ancient‑cipher score' (Galois representation). On the other hand, by performing harmonic analysis on these cohomology spaces, we can find a particular basis, precisely characterized by eigenvalues of **automorphic forms**—corresponding to that 'modern‑staff score' (automorphic form)."
"Thus, in this concrete geometric model," she concluded, her voice slightly raised with excitement, "we have realized the Langlands correspondence! We've found a common 'stage' (geometric space) where number theory's 'ancient chant' (Galois representation) and harmonic analysis's 'complex symphony' (automorphic form) are revealed, on this stage, as two different renditions of the same mathematical drama!"
The hall fell silent, then erupted into sustained, enthusiastic applause. Yue'er's talk not only clearly mapped a concrete geometric path toward realizing the Langlands correspondence; more importantly, she brought that seemingly remote grand program into a realm where mathematical tools could concretely operate and study it. She laid the first solid pier for that bridge connecting number theory and harmonic analysis.
During the following Q&A, she calmly handled sharp, penetrating questions from top scholars in the audience, impressing everyone with her agile thinking, rigorous logic, and grasp of the mathematical framework. Standing at the podium, she seemed enveloped in a radiance emanating from pure intellect—confident, serene, yet brimming with the power of a trailblazer.
Meanwhile, in a distant Shanghai, in the top‑floor office of "Guantao Capital," Mozi sat alone, repeatedly watching the recorded video of this talk. He turned off the trading screen's dancing numbers, devoting all attention to understanding the mysterious, magnificent mathematical world Yue'er presented.
He certainly couldn't fully grasp the complex mathematical details—ℓ‑adic cohomology, Shimura varieties, eigenvalues of automorphic representations... these terms were like hieroglyphs to him. Yet he clung tightly to Yue'er's exquisite "musical score" metaphor. He strove to comprehend that grand vision of translating and unifying two utterly different "mathematical languages."
He watched the confident, eloquent Yue'er on screen, seeing the light in her eyes for unrelenting truth‑seeking, watching her use simple, elegant analogies to unravel the deepest mathematical concepts. An indescribable emotion, like a deep‑sea undercurrent, slowly wrapped around his heart.
That wasn't merely academic admiration; it was a soul‑deep fascination. He was awestruck by her intelligence, captivated by the regal presence she displayed in her field. She was like a knight exploring the universe of abstract thought, cutting through thorns and brambles, paving paths to unknown realms for mortals like him. The light she emitted differed from Xiuxiu's grounded resilience and warmth—it was a clear, pure, dazzling starlight originating from the extreme of logic and imagination.
He repeatedly dragged the progress bar, replaying the segment where Yue'er explained "score translation." Though he couldn't fully understand, he could sense the tremendous effort and profound beauty in trying to grasp the underlying interconnectedness of the world. In spiritual essence, how similar this was to his own attempts to find order in financial‑market chaos, to Xiuxiu's efforts to break through physical limits!
He turned off the video, leaned back in his chair, silent for a long while. Outside was Shanghai's bustling nightscape, flowing car lights like torrents of data. Yet his mind lingered on Yue'er's figure at the podium, and that metaphor of "cosmic scores," brimming with infinite imagination. He realized his feelings for Yue'er had long transcended initial intellectual resonance; they'd grown deeper, more complex, mingling admiration, appreciation, and a profound yearning to understand and approach that brilliant starry sky. This emotion, so different yet equally real and intense as the reliance and warmth rooted in shared struggle he felt for Xiuxiu, existed like two stars in his inner cosmos—with distinct orbits yet equally bright.